Quantum Computing with Electron Spins in Semiconductor Quantum Dots

Transcription

1 Quantum Computing with Electron Spins in Semiconductor Quantum Dots Rajesh Poddar January 9, 7 Junior Paper submitted to the Department of Physics, Princeton University in partial fulfillment of the requirement for the degree of A.B. in Physics Advisor: Second Reader: William F. Brinkman Jason R. Petta Abstract Quantum computation has been the subject of a considerable research effort in the last decade. In this paper we discuss the Loss and DiVincenzo proposal [1] to implement a quantum computer in the solid state along with experimental advances in realizing the elements of this proposal. Loss and DiVincenzo propose the use of spin 1/ states of electrons confined to quantum dots. Here we present a detailed discussion of the physics of electron spins in quantum dots. We also describe the electromagnetic interactions of these spins with themselves and with external fields that can be used to realize quantum computation. Finally we review the experimental techniques used in initializing, controlling, characterizing, and measuring spins in few-electron quantum dots.

2 Contents 1 Introduction and Overview The Power of Quantum Computation 3 Quantum Dots: A Possible Implementation 4 4 Isolated Single Quantum Dots Fabrication of Quantum Dots The Constant Interaction model Quantum Mechanics of a Quantum Dot Spin States of a Single Electron in a Quantum Dot Spin States of Two Electrons in a Quantum Dot Coupled Quantum Dots The Constant Interaction Model for Coupled Quantum Dots Quantum Mechanics of Coupled Quantum Dots The Heisenberg Spin Hamiltonian The Exchange Integral Pauli Spin Blockade Coherent Manipulation of Single Spins Electron Spin Resonance (ESR) Experimental Realization of ESR in Quantum Dots Coherent Manipulation of Coupled Spins 8 Spin Readout Spin to Charge Conversion Spin to Charge Conversion using Difference in Energy of the Spin States Spin to Charge Conversion using Difference in Tunneling Rates of the Spin States Non-destructive Measurement of Spin Charge Measurement using QPCs Environmental Interactions T 1 and T Use of spin-echo to increase the coherence time of coupled dots Conclusion 9 11 Appendix I: The Constant Interaction Model for a Coupled Quantum Dot System 3 1 Appendix II: The Controlled NOT operation from the Heisenberg Spin Hamiltonian 31 References 33 1

3 1 Introduction and Overview Practical implementation of quantum computers has been a major research goal in the last few decades and is still a very active field. Various implementation schemes have been proposed and yet no single one has emerged as a clear winner. In this paper we focus on one particular implementation scheme originally proposed by Daniel Loss and David P. DiVincenzo in 1998 [1]. The proposed implementation uses the spin 1/ states of electrons confined in semiconductor quantum dots. These have been shown to have long coherence times [], a key ingredient for successful implementation of qubits. Since existing nanofabrication techniques have made creating quantum dots with desired properties a routine task, the proposal is very scalable. Many key elements of the proposal have already been realized experimentally [3]. Isolating single electrons in quantum dots, reading out the spin states of electrons [4, 5, 6], swapping the spins of two neighboring electrons in coupled quantum dots [] and rotating the spin of single electrons in a controlled fashion [3] have all been successfully demonstrated. These coherent manipulations of single and coupled spins can be used to realize universal quantum computation [7]. This paper is organized into several sections. In section, we briefly introduce what quantum computation is and give some examples to demonstrate how it can be more powerful than classical computation. The rest of the paper discusses the use of spin states of single-electron semiconductor quantum dots for quantum computation. Section 3 discusses the proposal of Loss and DiVincenzo to use quantum dots for quantum computation. In section 4, we discuss the classical and quantum physics of single isolated quantum dots. Section 5 moves on to the physics of a system of coupled dots. Section 6 discusses the theory and experimental realization of single spin rotations and Section 7 describes the coherent control of two spins in neighboring dots. We discuss spin readout using spin to charge conversion in Section 8. This section also contains a description of Quantum Point Contacts (QPC) used as electrometers. Finally Section 9 discusses the non-ideal behavior of the system and its environmental interactions. The Power of Quantum Computation A classical computer with an n-bit register can be in one of n states. An n-qubit quantum computer in contrast can exist in a normalized linear superposition of these n classical states. This is a vector space over the field of complex numbers and will be denoted by H n, the Hilbert space of n qubits. A classical n-bit computation corresponds to an arbitrary function f : {, 1} n {, 1} n. A quantum computation corresponds to unitary operator on H n ; U : H n H n s.t. UU = I where U is the conjugate transpose of U and I is the n n identity matrix. We denote the states of a quantum computer in Dirac bra-ket notation. Also, if x H A, y H B, then x y H A H B, the tensor product of spaces H A and H B. The tensor product is a formal product and is defined to be bilinear in both its operands. Therefore, Consider a quantum computation ( a + b ) ( c + d ) = a c + a d + b c + b d U : x x f(x), (1) where the Hilbert space that U operates on, is a tensor product of the input register and the output register. Since we can apply U to any vector in this Hilbert space, applying it to the following state of the input register x {,1} n x ()

4 produces x {,1} n x f(x). (3) A single unitary operation computes f(x) for all values of x {, 1} n. It is crucial to note that the value of f(x) for all x cannot be obtained by performing this computation only once, since a quantum mechanical measurement of this state will inevitably lead to some loss of information. However, by exploiting this massive parallelism cleverly, it is possible to sometimes gain an exponential advantage over classical computers [8]. A quantum algorithm for factoring large integers is known that results in an exponential speedup resulting in a polynomial time algorithm for this problem [9] with applications in cryptography. Also, simulation of quantum phenomenon in a classical computer takes time exponential in the number of qubits simulated. Quantum computers will allow extremely efficient simulation of quantum phenomenon. We give a simple example that demonstrates how quantum computation can be more powerful than classical computation. Suppose f(x) is a function that takes as input a single bit and outputs a single bit. We wish to determine whether f() = f(1). Classically, the only way to do this requires evaluating the function f at and 1 and comparing the results. However, only one unitary operation followed by a single measurement is sufficient to know whether f() = f(1) with a quantum computer. The quantum mechanical analogue of f needs to be invertible since quantum mechanics only allows unitary operations. However f may not necessarily be invertible. Hence we consider the following unitary transformation. U : x y x y f(x). (4) In this setup it is possible to know whether f() = f(1) with one operation U followed by a measurement. (We ignore the normalization in the following discussion.) U = f() (5) U 1 = f(1) (6) U 1 = f() (7) U 1 1 = f(1) (8) f(1) is the complement (1 is the complement of and vice versa) of f(1). Taking x = + 1 and y = 1 we get If f() = f(1) then the final state will be U x y = U[( + 1 )( 1 )] (9) If f() f(1) then f() = f(1) and the final state will be = U[ ] (1) = f() + 1 f(1) f() 1 f(1) (11) ( + 1 )( f() f() (1) ( 1 )( f() f() (13) Therefore measuring the first qubit after an application of U to ( + 1 )( 1 ) in the {( + 1 ), ( 1 )} basis unambiguously tells us if f() = f(1). The above discussion is adapted from [8] and it contains an excellent and thorough discussion of quantum computation. 3

5 3 Quantum Dots: A Possible Implementation Because of the interesting theoretical possibilities presented by a quantum computer, a practical realization has become a major research goal. Here we discuss one proposed implementation that uses the spin degrees of freedom of singleelectron quantum dots [1]. Quantum dots are semiconductor nano-structures in which the motion of electrons in the conduction band of the semiconductor are confined in all 3 dimensions [1, 11]. This is very similar to the confinement of electrons by the Coulomb attraction of nuclei. Hence, semiconductor quantum dots are also called artificial atoms [11]. With recent advances in nano-fabrication technology, it is now routine to create a lateral quantum dot device with the number of electrons in the dot precisely tunable to, 1,, or more, by applying electrostatic voltages to the gate structure defining the dot [1]. Furthermore, it is also possible to fabricate dots next to each other with electrostatically tunable coupling between them [13]. Quantum dot fabrication is discussed in more detail in the next section. Loss and DiVincenzo proposed a possible scalable realization of qubits in such coupled dot systems. In this proposal, a quantum computer is created by defining an array of lateral quantum dots in some semiconductor substrate. The quantum dots are tuned to the 1 electron regime. A qubit is the spin of the excess electron in a quantum dot. 1- qubit operations are performed by rotating the spins in these dots. -qubit gates are implemented by temporarily coupling the electrons in adjacent dots thereby allowing unitary time evolution described by the Hamiltonian of the coupled system. Qubits can be initialized by letting them relax to the ground state in a high magnetic field [1]. When the Zeeman splitting is much larger than the electron temperature, gµ B B k B T, almost all electrons will align themselves with the magnetic field if g is positive and anti-align otherwise. Because of the small magnetic moment of electron spins, the spin states do not couple strongly to the environment, thereby leading to long coherence times [14]. However, spin readout becomes a challenge precisely because of this small magnetic moment. One solution that has already been implemented is spin-to-charge conversion. As discussed in more detail in a later section, either the energy difference between the two spin states [4] or the difference in the rate of tunneling into the reservoir [5, 6] can be used to convert spin to charge. Reading out charge is much easier and can be done either via electron transport experiments or via the use of Quantum Point Contacts as electrometers. 1-qubit gates are necessary for universal quantum computation. In this case, 1-qubit gates correspond to single spin rotations and at least two mechanisms seem feasible. One mechanism is to use time varying Zeeman coupling between the spin and an external magnetic field. The Hamiltonian in this case is H(t) = g(t)µ B S B(t) (14) This Hamiltonian results in the spin rotating about the B(t) axis. The Hamiltonian can be controlled either by varying the magnetic field or the electron g factor. A spin that needs to be rotated can be moved to a different layer of the semiconductor substrate by applying electric potentials to the dot containing the spin. This different layer may have a different g factor and/or a different magnetic field. The other mechanism is to apply an oscillating magnetic field perpendicular to the direction of a static magnetic field leading to Electron Spin Resonance (ESR) via Rabi oscillations if the frequency of the oscillating magnetic field matches the Zeeman splitting of the spin. This mechanism has been experimentally demonstrated [3] and is discussed in detail later. For -qubit gates, coherent manipulation of qubits in the Loss and DiVincenzo proposal is done by electrically controlling the tunnel coupling between adjacent quantum dots. Due to zero dimensional confinement of electrons in quantum dots, the orbital energy spectrum of confined electrons is discrete. If E is the spacing between the ground state and the first excited state of a single dot, then for sufficiently low temperatures, k B T E, the orbital excited states are not accessible to the system. We show in Section 7 that the effective Hamiltonian of the two electrons in two 4

6 adjacent dots can be simplified to the Heisenberg spin Hamiltonian H(t) = J(t)S 1 S (15) where S i is the spin operator of electron i and J(t) is the energy difference between the singlet and triplet spin states of the -electron system. The origin of this Hamiltonian is not due to spin-spin interactions but due to Coulomb interactions between the electrons and the Pauli principle. The central idea is that J(t) is electrically controllable. For small coupling between the two dots J(t) is close to where as for significant coupling, J(t) is large. The Hamiltonian in Equation 15 can be used to build the quantum XOR gate if J(t) is just right. The quantum XOR gate along with single qubit rotations can be used to implement any unitary operator. The XOR gate is defined as: U XOR : x y x x y. (16) This gate in turn can be built using a simpler gate, Usw 1/, the square root of the swap operator. The swap operator simply exchanges the states of qubits. A detailed derivation of U XOR from the spin Hamiltonian is given in Appendix II. and single qubit rotations reliably, problems in non-equilibrium spin dynamics. Therefore, for the rest of the paper, we describe in detail the physics of spins in few electron quantum dots. The problem of building quantum computers is hence reduced to physically implementing U 1/ sw 4 Isolated Single Quantum Dots A quantum dot is an artificial structure in which electrons or holes are confined in all 3 dimensions. There exist various schemes to create quantum dots. Here, we focus on quantum dots created in the -dimensional electron gas (DEG) of semiconductor heterostructures. A DEG is a construct in which electrons are restricted from moving in one dimension but can freely move in the other two. 4.1 Fabrication of Quantum Dots One popular method to create a DEG is to make a semiconductor crystal containing AlGaAs on top of GaAs [1]. Such structures are usually created using molecular beam epitaxy which builds up the crystal one atomic layer at a time. Hence, the interface of GaAs and AlGaAs in the heterostructure is highly regular. AlGaAs is doped with Si to introduce free electrons in the DEG. Due to the difference in the band structures of the two semiconductors, these free electrons accumulate at the interface of AlGaAs and GaAs. More specifically, the edge of the conduction band of AlGaAs is higher than that of GaAs. Therefore, the free electrons from Si find it energetically favorable to move to GaAs. However, these free electrons do not penetrate deep into GaAs since they are attracted by the +ve charge they leave behind. Therefore, the edge of the conduction band of the heterostructure has a local minimum at the interface of AlGaAs and GaAs. By choosing the right amount of doping, the Fermi energy of the free electrons in the heterostructure can be made to lie very slightly above this dent. Hence, free electrons get trapped at the interface. Confinement in the other dimensions can be achieved in at least two different ways. When a round pillar is etched in the direction perpendicular to the DEG, a vertical quantum dot is created [11]. Confinement in the other two dimensions in vertical quantum dots is hence achieved by limiting the extent of the semiconductor itself. However, vertical quantum dots are not preferred for manipulating single electron spins since they do not readily allow for controlling the different parameters of a dot like its size and extent of coupling to source/drain electron reservoirs via tunnel barriers. 5

7 Figure 1: Schematic diagram of a lateral quantum dot on an Al- GaAs/GaAs DEG defined using metallic gates. Figure from [1]. Figure : Electron micrograph of a lateral quantum dot. The white circle is the dot, the light areas are the metallic gates and the crosses are points of ohmic contact which allow currents to flow through the dot. Figure adapted from [1]. Figure 3: The electrical network used to model a single quantum dot. Figure from [15]. In contrast, lateral quantum dots are defined by fabricating metallic gates on the top of the DEG on the surface of AlGaAs (Figure 1). The metallic gates are patterned so that they enclose a circular region of approximately.5µm (Figure ). Electron beam lithography allows the patterning of gates to a nanometer level precision. By applying negative voltages on these gates, the region directly below the gate is depleted of electrons thereby effectively defining a quantum dot at the center. The thick gates in Figure define the shape of the dot. The thin gate at the top of the Figure in combination with the left(right) gate allows tuning the size of the tunneling barrier between the dot and the left (right) reservoir. The thin gate at the bottom of the figure, is used to change the electrostatic potential of the dot. By making this potential sufficiently negative, the dot can be completely emptied of free electrons. One electron at a time can then be added to these dots by making the gate potential, V g less negative. Under appropriate conditions, a current can flow through the dot if a potential difference is created between the source and the drain, V sd [1]. This corresponds to applying V sd between the terminals marked by crosses in Figure. A Quantum Point Contact can be created near the dot to serve as an electrometer that measures the charge residing in the dot. QPCs are discussed in greater detail in section The Constant Interaction model Here we describe a very simple semi-classical model that can account for a variety of effects observed in quantum dots. In this model the Coulomb interactions of electrons in the dot amongst themselves and with the electrons on the gates that they are coupled to through tunneling barriers are described by various capacitances as in Figure 3. A more detailed discussion of this model can be found in [15]. The energy of the dot when there are N excess electrons in it can be calculated as follows. Q 1 = C L (V 1 V L ) + C R (V 1 V R ) + C G (V 1 V G ) (17) Q 1 + C L V L + C R V R + C G V G = C 1 V 1, C 1 = C L + C G + C R (18) Q 1 = e (N N ), where N is the number of electrons in the dot when all voltage sources are. N exists to 6

8 Figure 4: a. Coulomb Blockage mode: electrons cannot tunnel through the dot. b. Coulomb Blockade is lifted. c. Gate Voltage vs. Current through the dot. Figure from [1]. compensate the background positive charge of the Si donors in the AlGaAs layer of the heterostructure. U(N) = (C 1V 1 ) C 1 + N E i (19) i=1 = ( e (N N ) + C L V L + C R V R + C G V G ) C 1 + N E i () Due to confinement, the energy spectrum of single particle states is discretized. Here, E i corresponds to the eigenvalue of the ith single particle energy eigenstate. Therefore, the expression in equation corresponds to the energy when all N electrons are in the ground state. Appropriate adjustments need to be made when some electrons are in the excited state. We will compute E i under some reasonable approximations in the next subsection. The electrochemical potential µ(n) of the transition from the ground state of N 1 electrons to the ground state of N electrons is given by µ(n) = U(N) U(N 1) (1) The spacing between the electrochemical potentials is given by = (N N 1 )E C E C e (C LV LR + C G V G ) + E N. () E add = µ(n) µ(n 1) = E C + E, E = E N E N 1 (3) Here E C = e /C 1 is known as the charging energy [1]. It corresponds to electrostatic contribution of the amount of extra energy needed to add another electron to the dot. E can be thought of as the quantum mechanical contribution to the addition energy. Quantum dots display a distinctive phenomenon called Coulomb blockade which can be understood with E add. An electron can tunnel through the dot only if the electrochemical potential of next available state, which is E add above the current state, is between the electrochemical potentials of the source and the drain (see Figure 4). Otherwise, the dot is in the Coulomb blockade mode and hence no current can flow through the dot. Therefore, by sweeping i=1 7

9 the gate voltage from very negative to less negative and measuring the current during this procedure, one obtains the graph in Figure 4c. The peaks in the graph correspond to the value of gate voltages for which the the electrochemical potential of the next available state lies within the bias window. The bias window is the range of potentials between the potentials of the source and the drain. The region between the peaks corresponds to states of the dot with a fixed number of electrons, i.e. the Coulomb Blockade mode. 4.3 Quantum Mechanics of a Quantum Dot Now, we derive the energy spectrum of a single electron in a quantum dot. We will assume that there is a magnetic field of magnitude B perpendicular to the plane of the DEG in which the quantum dot resides. Such a magnetic field is usually present in experiments that attempt to coherently manipulate the spins in few electron quantum dots. Define the x and y axes to be in the plane of the DEG and the z axis to be perpendicular the the DEG. We will assume that the confining potential in the z axis is the box potential and the confining potential in the x y plane is the rotationally symmetric parabolic potential. The Hamiltonian is V (z) = For B along the z-axis, B = (,, B) we choose the vector potential This gives H = 1 m (p + e c A) + V (x, y) + V (z) (4) { }, z < a, V (x, y) = mω otherwise (x + y ) (5) A = ( By/, Bx/, ) (6) H = 1 m (p x + p y) + m(ω + ω) (x + y ) + Ω(xp y yp x ) + p z + V (z) (7) m with Ω = eb mc. This separates into Hamiltonians H = Hxy + Hz (8) H z is just the particle in a 1 dimensional box Hamiltonian and hence has eigenenergies E z,n = π n 8ma (9) Because of the extremely small value of a in a DEG, only the ground state will be accessible at the temperatures of interest. Consequently we will ignore the z direction completely from now on. For H xy let ω = Ω + ω. Then H xy = 1 m (p x + p y) + mω (x + y ) + Ω(xp y yp x ). (3) Now, we introduce the SHO creation and annihilation operators: mω 1 a 1 = x + i m ω p x, a = mω 1 y + i m ω p y (31) 8

10 Figure 5: Energy spectrum of an electron in dimensions with circularly symmetric parabolic confinement potential and a static magnetic field. Figure from [16]. a mω 1 1 = x i m ω p x, a mω 1 = y i m ω p y (3) These satisfy the commutation relationship [a i, a j ] = δ ij. With these, H xy = ω (a 1 a 1 + a a + 1) + i Ω(a a 1 a 1 a ) (33) Finally we perform another set of transformations that preserves the commutation relationship [b i, b j ] = δ ij. With this, the Hamiltonian simplifies to Finally, with quantum numbers n i we have b 1 = 1 (a 1 + ia ), b 1 = 1 (a 1 ia ) (34) b = 1 (ia 1 + a ), b 1 = 1 ( ia 1 + a ) (35) H xy = (ω + Ω )(b 1 b 1 + 1) + (ω Ω)(b b + 1) (36) b i b i n i = n i n i. (37) Therefore, the energy spectrum of a single electron in a quantum dot in the presence of a static magnetic field perpendicular to the direction of the DEG is E = ω + Ω (n 1 + n + 1) + Ω(n 1 n ) (38) This result is plotted in Figure 5. Solutions to elliptic confinement potential can be found in [16]. 9

11 Figure 6: Results of spin transport experiment described in the text. Plots a-f show the differential conductance of the dot as a function of V sd, the source-drain voltage and V G, the gate voltage. The 4 lines in figures a and b correspond to the 4 energy levels of a single electron quantum dot. From bottom up, they are orbital GS with spin up, orbital GS with spin down, orbital first ES with spin-up and orbital first ES with spin down. The rightmost plot shows fitting the value of the Zeeman splitting to the data in a-f. Figure from [1] which in turn is adapted from [17, 18]. 4.4 Spin States of a Single Electron in a Quantum Dot Having demonstrated that the eigenspectrum is discrete, we now consider the spin states of the system. For a single spin 1/ electron there are only two spin eigenstates in any given direction. Therefore considering the two lowest orbital eigenstates results in a 4 state system [1]. Let E ( ),G(E) correspond to the energy of the electron ground (first excited) state with the spin pointing in the +z ( z) direction. Then E,G = E,G + E Z (39) E,E = E,G + E orb (4) E,E = E,G + E orb + E Z (41) Here, E Z = gµ B B is the Zeeman energy. E orb can be found from Eq. 38 by setting n 1 = n = for the ground state and n 1 =, n = 1 for the first excited state. ( ) eb E orb = (ω Ω) = (ω 1 + eb mω c mc ) (43) Both E Z and E orb are measurable through electron transport experiments and hence can let us determine the values of g and ω. In such experiments, the variables that are independently controlled are the source drain voltage, V SD, and the dot gate voltage, V G. The measured variable is the current through the dot, I dot. Due to Coulomb Blockade, current can only flow through the dot when V G is such that the electrochemical potential of the next available state in the dot lies within the window set by V D. The plots in Figure 6 shows the results of an actual experiment [1, 17, 18]. Each plot in the figure has the differential conductance of the dot, di dot /dv SD, plotted in gray scale. Dark lines in the plot corresponds to transitions between Coulomb Blockade mode and the lifting of Coulomb Blockade. For V SD close to, the dot can conduct electrons only for very specific values of V G, since V G must be set so that electrochemical potential of a 1 electron state lies within the narrow window exposed by V SD. Similarly for large values of V SD, the dot can conduct for a range of values of V G. This is the reason for the V shaped regions in all these plots. Furthermore, if more than one state is available for an electron as it tunnels through the dot, the current is higher. The results depicted in Fig. 6 shows the plots for the -1 electron transition in the quantum dot. As expected, we see 4 dark lines in the plots to the left of (4) 1

12 the figure corresponding to the 4 states available to an electron. The other plots in the figure only show the first two lines. The vertical distance between these lines, corresponds to E Z and the distance between the first and third lines corresponds to E orb. Each plot has an inset describing the magnetic field used in the experiment. As expected, increasing the magnetic field increases the Zeeman splitting. 4.5 Spin States of Two Electrons in a Quantum Dot We again consider the 4 lowest electron states. Since we have spin 1/ electrons now, the spin state space is 4 dimensional. The individual spins are no longer good quantum numbers, since in the presence of a magnetic field, the eigenstates of energy are the eigenstates of the total spin. The spin eigenstates are: S = 1 ( (44) T + = (45) T = 1 ( + (46) T = (47) By the Pauli principle, the total wavefunction must be anti-symmetric. Since the S state is anti-symmetric, both electrons can occupy the orbital ground states. However, the T states are symmetric, so their orbital part must be antisymmetric. Therefore, in the Heitler - London approximation, the orbital part will be an antisymmetric combination of the ground state, G, and the first excited state, E, 1 ( G E E G ) (48) To a first approximation, the orbital ground state energy is E and the first excited state energy is E + E 1 = E + E orb. Here E and E 1 are the single electron orbital ground and first excited state energies respectively and E orb is the difference between the two. This approximation ignores the repulsion between electrons and hence, the actual energies are higher for both states. We can correct for Coulomb interactions by introducing an energy term E K, that modifies E orb. We define the actual difference between the singlet and triplet states to be E ST. We note that E ST < E orb. Intuitively, this is because, both electrons cannot occupy the same position in the antisymmetric orbital state due to the Pauli principle. Therefore, the average distance between the two electrons in the spin triplet state is higher than that for the spin singlet state. Consequently the net Coulomb repulsion energy is lower for the triplet state than the singlet state. Therefore, E K serves to lower the energy difference between the two lowest states as calculated by the first approximation. Hence, E ST = E orb E K [1]. Finally, we remark that E K is similar to but not the same as the exchange energy to be discussed in Section 5.. With coupled quantum dots, E K serves to make the triplet state energetically less favorable since we have to account for the possibility of the electron hopping between the two dots. Following the footsteps of the constant interaction model, and the above discussion, we write down the energies of each of the electron states. U S = E,G + E,G + E C = E,G + E Z + E C (49) U T+ = E,G + E,E + E C E K = E,G + E ST + E C (5) U T = E,G + E,E + E C E K = E,G + E ST + E Z + E C (51) U T = E,G + E,E + E C E K = E,G + E Z + E ST + E C (5) 11

13 Figure 7: Spin transport experiment of a quantum dot at the 1 electron transition. A. Energy levels of various states involved in the transition. B. Electrochemical potential of the transitions. C. Expected differential conductance due to these transitions D. Experimental data which resembles C very closely. Figure from [1] which in turn is adapted from [19]. 1

14 Figure 8: Electron micrograph of a coupled quantum dot system. Figure from [13]. Figure 9: Charge Stability Diagram for a double dot system obtained experimentally using QPC. Figure from []. Using the same experimental setup as described earlier, it is possible to measure E ST. This combined with the the results of the the single spin transport measurements gives an experimentally measured value of E K. To measure E ST, the same experiment as the one in Figure 6 is conducted, except that the gate voltage is varied in the regime of transition between 1 and electrons. The result of such an experiment is shown in Figure 7d. In the figure, CB refers to Coulomb Blockade and N is the number of electrons in the dot. When N = 1, there is either a spin-up ( ) or spin-down ( ) electron in the quantum dot. The regions A-F corresponds to the lifting of Coulomb Blockade, i.e. electrons can tunnel through the dot via one or more of the S, T, T + and T states. Part a of the figure shows the the energy levels of 1 electron and electron states for a single quantum dot. Part b shows the electrochemical potentials of these transitions, i.e. the energy difference between the states involved in the transition. Part c shows a schematic of the expected results and d shows the actual results which agrees very well with the schematic. As can be seen from parts b and c, the vertical distance between various lines in part d can be used to calculate E Z and E ST. 5 Coupled Quantum Dots Now we consider the case of two lateral quantum dots next to each other with tunable coupling between the two dots. Figure 8 shows an electron micrograph of one such system [13]. The gates L and T allow setting the height of the tunneling barrier between the left dot and the left reservoir. Similarly the gates R and T allow tuning the tunneling barrier between the right dot and the right reservoir. Gates T and M set the coupling between the dots. The plunger gates, P L and P R, allow changing the electrostatic potential of the left and right dot respectively. There is a QPC associated with each dot. The QPC is used to detect the number of electrons in the dots. By applying a negative voltage to the QPC-L and L (QPC-R and R) gates, the current flowing from Source1 to Drain1 (Source to Drain) has to pass through the very narrow constriction between the gates QPC-L and L (QPC-R and R). The conductance of this narrow channel is highly susceptible to the number of electrons in the quantum dots. As the number of electrons in the dots increases, Coulomb repulsion due to these electrons affects the current passing through the QPC, thereby reducing its conductance. As is evident, this is a highly customizable coupled quantum dot design and hence has led to a number of experimental advances in coherent manipulation of electron spins. 13

15 Figure 1: Stable charge configurations as functions of the plunger gate voltages. a) is for the case when C M = and b) is for C M >. Figure from [15]. Figure 11: Schematic positions of the electrochemical potentials of the quantum dots, source and drain for the dotted box in Figure 1. Figure from [15]. 5.1 The Constant Interaction Model for Coupled Quantum Dots In analogy with the calculation for single quantum dot devices, the energies of electrons in a double dot device can also be calculated under the Constant Interaction Model. This is done in Appendix I. The electrochemical potentials are [15]: µ 1 (N 1, N ) = U(N 1, N ) U(N 1 1, N ) = (N 1 1 )E C 1 + N E CM E C 1 e (C G1V G1 ) E C M (C G V G ) + E 1,N e µ (N 1, N ) = U(N 1, N ) U(N 1, N 1) = (N 1 )E C + N 1 E CM E C e (C GV G ) E C M e (C G1 V G1 ) + E,N Since we have set V L = V R =, for fixed values of the capacitances and gate voltages, the largest integers N 1 and N such that µ 1 and µ are less than is the stable charge configuration of the coupled dot system. Figure 1 shows the partition of the dot potential space into regions of stable charge configurations. At the boundaries of the stable configurations, the electrochemical potential of the next available state is exactly (the electrochemical potential of the source and drain). Hence the double dot system can conduct current. In the interior of the stability patches, the dots are in Coulomb blockade mode. For instance, consider the triple points in the boxed region of Figure 1b. When the system is in the lower triple point, current can flow through the dot via the dot configurations (, ) (, 1) (1, ) (, ). ((m, n) refers to m electrons in the left dot and n in the right. Also, the source is on the left and the drain is on the right.) Therefore, electrons flow from right to left. For the upper triple point, current is conducted by holes. Hence the dot state sequence is (1, 1) (, 1) (1, ) (, ). When the dot potentials 14

16 are set such that the system lies exactly on the line between the two triple points, then an electron can go back and forth between the two dots but cannot go to either reservoir. Figure 11 explains this diagrammatically. Figure 9 shows experimentally determined charge stability diagram using a QPC. In this figure, color represents the conductance of a nearby QPC. 5. Quantum Mechanics of Coupled Quantum Dots We remarked earlier that the low temperature behavior of a coupled quantum dot system can be described by the Heisenberg Spin Hamiltonian: H(t) = J(t)S 1 S (53) Here S i is the spin operator of electron i and J(t) is coupling between the spins. Now, we elaborate on how Coulomb interactions between electrons combined with the Pauli Exclusion principle for electrons leads to this form of Hamiltonian at sufficiently low temperatures The Heisenberg Spin Hamiltonian A remarkable effect of the Pauli principle is that even in a Hamiltonian independent of spin terms, spin-dependent eigenstates arise [, 1, ]. Spin-orbit and spin-spin interactions are very small in quantum dots due to the small magnetic moment of electrons and hence can be safely ignored in this discussion. Hence the Hamiltonian of a coupled dot system is manifestly spin independent. Consequently, every quantum state of the coupled dot system can be expressed as a tensor product of the orbital and spin states: ψ = φ(r 1, r ) X (54) Here, φ is the orbital part of the state, r i indexes the position of the i th electron and X is an arbitrary spin state of the coupled system, i.e., a linear combination of S, T, T and T + states defined in Equations By the Pauli Principle, the total wavefunction ψ must be antisymmetric under exchange of both spin and spatial coordinates. Therefore, an energy eigenstate with a total spin of, i.e. the singlet state which is anti-symmetric, must have an orbital part that is symmetric under exchange. Similarly, energy eigenstates with a total spin of 1 must have an antisymmetric orbital part. Therefore, even though the Hamiltonian is spin-independent, the symmetry of the energy eigenstates depends on the total spin of the eigenstate []. Suppose φ S (r 1, r ) and φ T (r 1, r ) are the lowest energy symmetric and anti-symmetric orbital eigenstates of the coupled quantum dot system, with eigenenergies E S and E T respectively. For sufficiently low, temperatures, the Hilbert space of the coupled dot system is spanned by the following states: ψ S = φ S (r 1, r ) S (55) ψ T = φ T (r 1, r ) T (56) ψ T = φ T (r 1, r ) T (57) ψ T+ = φ T (r 1, r ) T + (58) The energy of ψ S is E S and that of the rest is E T. Next we note that for a spin 1/ electron S i = 1 ( 1 + 1) = 3 4. Therefore, S = (S 1 + S ) = 3 + S 1 S (59) 15

17 Figure 1: The Model used for the coupled dot device. Figure from [, 1] Also, S = for S and for the triplet states. Consequently, S 1 S has eigenvalue 1 4 for the triplet states and 3 4 for the singlet state. Therefore, the Hamiltonian H = 1 4 (E S + 3E T ) (E S E T )S 1 S (6) has eigenenergy E S for the singlet state and E T for the triplet state. Finally by redefining the zero of energy we obtain the Heisenberg Spin Hamiltonian: H = JS 1 S, J = E T E S (61) Next we describe the physical origin of J, the exchange term and present results from an approximate calculation of J using variational methods. 5.. The Exchange Integral We use the model sketched in Figure 1 for the coupled dot device. The quantum dots each have 1 electron confined to move in only the x-y plane. There is a static magnetic field, B, in the z direction and an electric field E in the x direction. Each electron experiences a circularly symmetric parabolic confinement potential. This is modeled using a quartic potential that separates into quadratic potentials at the dots. The individual dots are centered at ±a. The Hamiltonian of this system is: H = h i + C + H Z = H orb + H Z (6) i=1, 1 h i = m (p i e c A(r i)) + eex i + V (x, y) (63) e C = (64) κ r 1 r ( ) V (x, y) = mω 1 4a (x a ) + y (65) To calculate J we need to calculate the lowest energy symmetric and antisymmetric solutions to the Hamiltonian in Equation 6. However, this Hamiltonian is extremely complicated to solve exactly. Instead, we guess approximate solutions to the eigenstate problem of H. Let φ S and φ T be the symmetric and antisymmetric solutions respectively. Then the expected energy of these eigenstates can be used to obtain the exchange term. J = E T E S = φ T H φ T φ S H φ S (66) 16

18 Heitler-London Approximation In this approximation, φ S and φ T are taken to be normalized linear combinations of single electron ground state orbitals localized at each dot. We calculated the single electron states in section 4.3. As was demonstrated, the system could be decomposed into uncoupled harmonic oscillators. The ground state energy was calculated to be (. eb ω = ω 1 + mω c) The ground state wavefunction for a dot at the origin is mω ψ(x, y) = π e mω(x +y )/ (67) Translation of the entire system to (±a, ) produces mω ψ±(x, y) = π e mω((x±a) +y )/ Finally, we need to correct for the fact that magnetic field is shifted back to the origin. This involves a gauge transformation of A ± = ( By, B(x a), )/ to A ± = ( By, Bx, )/ which introduces a phase factor of e B ±iya l, where c l B is the magnetic length eb []. Therefore, the final single particle states for the coupled dot system are: ±iya l mω ψ ± (x, y) = e B π e mω((x±a) +y )/ The Heitler-London approximation for the symmetric ground state of the coupled dot is: φ S = ψ +(r 1 )ψ (r ) + ψ + (r )ψ (r 1 ), S = ψ + ψ = d rψ+(r)ψ (r) (7) (1 + S ) The first term in the numerator corresponds to electron 1 localized at the right quantum dot and electron localized at the left quantum dot. Similarly the second term corresponds to electron localized at the right quantum dot and electron 1 localized at the left quantum dot. The denominator is necessary to properly normalize the new wavefunction since the two single electron states are not necessarily orthonormal. Thus, under this approximation, φ T = ψ +(r 1 )ψ (r ) ψ + (r )ψ (r 1 ) (1 S ) Calculation of J using equations 66, 69, 7, and 71 is a straightforward though tedious task. The result obtained is [, 3, 1] ω J(B, E, d) = sinh (d [b 1 + Y + Z) (7) b ])(X X = c ( ) b e bd I (bd ) e d (b 1/b) I (d [b 1/b]) (73) Y = 3 4b (1 + bd ) (74) (68) (69) (71) Z = 3 d ( ) eea (75) ω Here, B is the strength of the magnetic field, E is the strength of the electric field, b = ω ω, d = a mω and I is the zeroth order Bessel function. The exchange energy is plotted for various values of the parameters J, B and d in Figures 13, 14 and 16. One robust observation from these plots is the consistent zero crossing of J. Therefore, at sufficiently large magnetic field, electric field or inter-dot distance, the triplet state becomes the ground state. 17

19 Figure 13: Exchange Energy, J, vs. Magnetic Field for different values of the Electric Field. Figure from [3]. Figure 14: Exchange Energy, J, vs. Electric Field for different values of the Magnetic Field. Figure from [3]. 5.3 Pauli Spin Blockade In addition to Coulomb Blockade, coupled quantum dots in the few-electron regime also demonstrate a feature known as Pauli Spin Blockade [1]. This is similar to Coulomb Blockade in that current flow through the dot is not possible during Spin Blockade. However, the origin of this phenomenon is the Pauli exclusion principle instead of Coulomb repulsion between electrons. Consider a double dot system with the gate, source, and drain voltages set as in Figure 15. Here, the dot on the right is such that there is always one electron in it. Another electron can be added to the right dot to form either the spin singlet or one of the triplet states. The electrochemical potential of the spin singlet state is shown in the figure. The triplet states in the right dot are above the singlet state and also above the 1 electron state in the left dot. The reservoir on the right is tuned such that if a second electron arrives at the right dot, it can easily tunnel out. Similarly the left dot and reservoirs are set up so that 1 electron can tunnel from the left reservoir to the left dot. Now suppose the electron in the left dot has a different spin than that of the right dot. In this case, the electron on the left dot can tunnel into the singlet state of the right dot and then out into the right reservoir. However, if the orientation of both spins is the same, then by the Pauli principle and conservation of total spin, the electron on the left dot can only tunnel into the triplet state of the right dot. Since that state is energetically inaccessible to the electron on the left dot, the electrons get stuck in their individual dots. This is known as spin blockade. 6 Coherent Manipulation of Single Spins As pointed out in an earlier section, single spin rotations are essential for quantum computation. For instance, the implementation of U XOR using Usw 1/ requires the ability to rotate individual spins. The Hamiltonian of a spin in a static magnetic field is the Zeeman Hamiltonian give by H = gµ B S B, S = 1 σ. (76) 18

20 Figure 15: Illustration of Pauli Spin Blockade. Figure from [1]. Figure 16: Exchange Energy, J, vs. normalized separation of dots for different values of the Magnetic Field. Figure from [3]. Here g is the electron g factor and µ B is the Bohr magneton. σ = (σ 1, σ, σ 3 ) are the Pauli spin matrices given by [ ] [ ] [ ] 1 i 1 σ 1 =, σ 1 =, σ i 3 = 1 Choosing the z-axis such that B = (,, B), results in the energy eigenspectrum [ ] 1 =, E = 1 [ ] gµ BB, =, E 1 = 1 gµ BB Hence the energy difference between the two states is gµ B B. Spin rotations can be performed by changing the Hamiltonian to introduce a coupling between these spin states. In principle this can be achieved by various techniques like spin-orbit coupling or Electron Spin Resonance (ESR). 6.1 Electron Spin Resonance (ESR) ESR involves the application of an oscillating magnetic field (B ac ) in a direction perpendicular to the static magnetic field (B ext ). If the frequency of the oscillating f is resonant to the spin precession frequency, i.e. hf ac = gµ B B ext then the electron spin oscillates between the and states. These oscillations are called Rabi oscillations. The Hamiltonian of an electron spin in the presence of both the static and oscillating magnetic field is In matrix form, this is H(t) = 1 gµ BB ext σ gµ BB ac cos ωtσ 1 (77) H(t) = 1 gµ B [ ] B ext B ac cos ωt B ac cos ωt B ext = 1 gµ BM (78) This Hamiltonian is only solvable in terms of elliptic integrals. Hence we will solve it exactly under the rotating wave approximation. When ω is resonant with the spin precession frequency due to B ext, we can invoke the following 19

21 Figure 17: The Bloch Sphere. Figure from [4]. Figure 18: Experimental setup for observing ESR in a coupled quantum dot system. Figure from [3]. approximation The Hamiltonian then becomes M 1 = B ac e iωt H(t) = 1 gµ B, M e iωt 1 = B ac [ B ext B ac e iωt B ac e iωt B ext This Hamiltonian can be solved exactly. The Schroedinger equation is: [ ] [ d a = i dt b gµ B ext B B ac e iωt B ac e iωt B ext The resonance condition implies that ω = gµ B B ext. Therefore, [ ] d a = iω [ 1 gµ dt b B B ac gµ B B ac ω eiωt ω e iωt 1 ] ] [ a b ] ] [ a b ] (79) (8) (81) (8) With the substitutions u = ae iωt/, v = be iωt/ we get [ ] d u = igµ BB ac dt v 4 [ 1 1 ] [ u v ] (83) With these substitutions, we move from the lab frame to the rotating frame in which there is no rotation around the static magnetic field. For initial state spin up, u() = 1 and v() =, and the solutions are ( ) ( ) gµb B ac gµb B ac u(t) = cos, v(t) = sin (84) 4 4 Therefore the probability of finding spin up at some later time t is a = u, which oscillates with frequency f rabi = gµ BB ac. (85) h

22 A Bloch sphere is extremely useful in visualizing this phenomenon. Consider an arbitrary quantum state of the single electron spin. We can parametrize it in terms of angular coordinates θ and φ such that [ ] [ ] a cos θ = b e iφ, θ < π, φ < π. (86) sin θ Then an arbitrary state of the system can be thought of as a vector (Bloch vector) attached to the origin with its head on the surface of a sphere (Bloch sphere) as in Figure 17. The North pole corresponds to spin up and the South Pole corresponds to spin down. With this picture, Rabi oscillations can be thought of as the Bloch vector spiraling on the Bloch sphere as in Figure Experimental Realization of ESR in Quantum Dots Even though the idea of Rabi oscillations is very simple, it is difficult to implement experimentally. Firstly, since the magnetic moment of an electron is very small, an RF pulse in the microwave regime is needed for the oscillating magnetic field. Since the RF pulse also includes an oscillating electric field, this interferes with the orbital degrees of freedom of the electron in the quantum dot. Previously, it had been observed that Coulomb blockade gets lifted for such high frequencies and so it becomes extremely difficult to observe Rabi oscillations. Koppens et. al. [3] used a coupled quantum dot system in the Pauli spin blockade mode to circumvent this problem. There is a large magnetic field perpendicular to the DEG and a coplanar stripline is fabricated on the dot to produce RF pulses. Note that the RF pulse applies to both dots. A simplified Hamiltonian of the system when there is no coupling between the dots is: H = H 1 + H, H i = gµ B S i (B ext + B i,n ) + gµ B cos (ωt)s i B ac (87) B 1(),N is the effective magnetic field on the left(right) dot due the spins of the the nuclei in the GaAs substrate that the DEG is in. This magnetic field is approximately 1mT in magnitude. For timescales involved in this experiment, these random magnetic fields can be considered to constant over time. An RF pulse of frequency f RF will induce Rabi oscillations in spin i if hf RF = gµ B B ext + B i,n. (88) Their experimental setup is illustrated in Figure 18. Initially, the double dot system is set up in the Pauli spin blockade mode described earlier. The gate potentials of double dot system are set such the the right dot always has one electron in it and if a second electron reaches the right dot then it can tunnel out to the reservoir. The potential of the reservoir coupled to the left dot is higher than the potential of the first electron in the left dot. Hence an electron can tunnel into the left dot. If the spins of the two electrons are different, then the electron on the left dot can tunnel into the singlet state of the right dot and subsequently tunnel out of the right dot. However, if the spins of the two electrons are the same, then the left electron has to tunnel to the triplet state of the right dot, which is energetically inaccessible. Therefore, initially both dots have 1 electron each whose spins are aligned. Then the gate voltage of the right dot is made more negative making both the triplet and the singlet states of the second dot energetically inaccessible. Hence no tunneling can happen and the system is in a stable (1,1) charge configuration in Coulomb blockade mode with aligned spins. Next, an RF burst of variable duration is applied to double dot system. This rotates the spin on the dots, only if the resonance condition in equation 88 is satisfied. At the end of the RF burst, the right dot potential is moved to the spin blockade mode. This projects the double dot system to the S(,) state ( electrons in singlet configuration in the right dot). If at the end of the RF pulse, the two spins are aligned anti-parallel to each other then the left electron can tunnel into the right dot. A tunneling even is detected as a current through the dot, I dot. 1

23 Figure 19: Rabi Oscillations on a Bloch Sphere. The green line represents the path of the Bloch vector during an oscillation. The red sinusoid is the z component of the spin vs time. Figure adapted from [5]. Figure : Experimentally observed Rabi Oscillations. Figure from [3]. Suppose that only the left dot is in resonance with RF pulse. Then the time evolution of the spin states when the RF pulse is applied will be +. Therefore the frequency of oscillation of I d ot will be the Rabi frequency as in equation 85. However, if both dots are in resonance with the RF pulse, then the time evolution will be + +. Note that the probability of tunneling is proportional to the probability that the spins are anti-aligned, i.e. the z component of the total spin S 1 + S is zero. In the nd and 4th step of sequence depicted above, the total spin is with probability half. Hence the expected frequency of tunneling is twice the rabi frequency. Also, the amplitude of the current is half as much the previous because the probability of the total spin being is only half. The oscillation of I dot was experimentally observed by Koppens et at and is reproduced in Figure. As expected, the period of oscillations changes linearly with the magnitude of the RF field that depends on the power supplied to the coplanar stripline. Only the low frequency oscillation corresponding to 1 electron in resonance with the magnetic field was observed in [3]. 7 Coherent Manipulation of Coupled Spins Another key ingredient for quantum computation is the ability to coherently manipulate qubits. As mentioned earlier, the quantum XOR gate is a key ingredient in quantum computation. Also, it was shown that this gate can in turn be built by composing single spin rotations and the square root of swap Usw 1/ gate. In this section, we describe an experiment that successfully demonstrated Usw 1/ [].

24 Figure 1: The effect of different values of the detuning parameter on the double dot system. Figure from []. Figure : Coherent control of coupled spins. A Experimental setup. B. Probability of the spin being in (, )S and the end of one cycle vs. detuning and exchange time. C and D are slices of B for fixed ɛ. Figure from []. requires the ability to control the exchange term, J in the Heisenberg spin Hamiltonian in Equation 15. Petta et al. achieve this by varying the detuning parameter, ɛ which is proportional to V L V R. V L controls the tunneling from the left reservoir to the left dot and V R controls the tunneling from the right dot to the right reservoir. The effect of the detuning parameter on the the double dot system and J is illustrated in Figure 1. The detuning parameter controls the relative energies of the (, ) and (1, 1) charge states of the double dot system. For ɛ > having an electron in the left dot is extremely expensive in energy terms and hence (, ) is the ground state of the system. This is how the system is initialized. Furthermore, the (, )T triplet state is about 4 µev above the (, )S singlet state and hence at the temperatures used in this experiment, initialization puts the system in (, )S. Similarly, for ɛ < the (1, 1) charge states become the ground state. For ɛ close to, the exchange interaction between the two dots, J is large due to the strong coupling between the (, ) and (1, 1) states. For ɛ, J effectively vanishes due to no coupling between the dots. Petta et al demonstrate oscillations in the, basis. Here, spin-up corresponds to alignment of the spin with the direction of the effective magnetic field experienced by the electrons in the double dot due to the nuclear spins in the semiconductor heterostructure. Half a period of this oscillation corresponds to U sw and a quarter period Implementing U sw or U 1/ sw corresponds to U 1/ sw. Their experiment and data is shown in Figure. After initialization into the (, )S state, ɛ is quickly reduced to a -ve value so that the electrons get separated into the dots. Then the detuning parameter is slowly lowered even further to the region where the (1, 1)S and the (1, 1)T states are degenerate and hence mix by the random nuclear field. Consequently the system ends up in the ground state of the nuclear field.. After this a pulse of variable duration τ E is applied to ɛ. This makes ɛ close to for time τ E effectively turning J on. Dynamics of the Heisenberg spin Hamiltonian takes place, rotating the spin in the, basis. Readout follows the same steps in reverse as illustrated in the figure. At the end of the cycle, the probability of ending up in (, )S states oscillates with τ E. 3

25 Figure 3: Experimental setup for measuring spins using using difference in energy between spin states. Figure from [4]. Figure 4: Typical data collected during the experiment in Figure 3. Figure from [4]. Part B of the figure shows in color, the probability of observing (, )S state at the end of each cycle as a function of ɛ and τ E. As expected, this oscillates with τ E. Furthermore, making ɛ close to increases J, which makes the oscillations faster. Theory suggests that half an oscillation, i.e. rotation by an angle of π, should take time J(ɛ)τ E as was derived in Equation 13. Parts C and D show slices of the data in B at fixed values of the detuning parameter. Both U sw and Usw 1/ can clearly be seen from the data. 8 Spin Readout For a quantum computer to be useful, it must be able to output the results of its computations. For the proposal in consideration here, this requires an ability to measure the value of the spin along some direction. However, since the magnetic moment of a single spin is extremely small, spin readout is a difficult problem. The solution that has been employed in recent experiments is a -step process. First, the spin is converted to charge and then the charge is measured either via a QPC or through the current generated by the charge. 8.1 Spin to Charge Conversion A destructive measurement is one in which, repeated measurements of the same spin does not give the same value [6]. This happens if the process of measurement leads to an evolution of the spin even after the measurement is completed. We first outline two mechanisms for measuring spin destructively, then we present a method to measure a spin non-destructively. 4

26 Figure 5: Experimental setup for measuring spin using difference in tunneling rates. Figure from [5]. Figure 6: Typical data collected during the experiment in Figure 5. Figure from [5] Spin to Charge Conversion using Difference in Energy of the Spin States Due to the Zeeman splitting between the spin states of an electron in the presence of a magnetic field, the energy of these states is different. This can be utilized in measuring the spin by tuning the barrier to the tunnel reservoir s.t. the spin up state is below the electrochemical potential of the reservoir whereas the spin down state is above the electrochemical potential. Hence, only the spin-down state can tunnel into the reservoir. Soon after the spin-down electron tunnels out of the dot, another spin up electron tunnels into the dot. Therefore, spin-down is correlated with electrons in the dot for a short period of time depending of the tunneling rate between the dot and the reservoir. Similarly spin-up is correlated with always 1 electron in the dot. The experimental setup is illustrated in Figure 3 and uses a single dot device with a nearby QPC [4]. Initially the single dot device is emptied of electrons by tuning the gate voltage such that both spins can tunnel out of the reservoir. Then, an electron is injected into the dot by moving the dot to the N = 1 Coulomb Blockade mode. The dot is then left to evolve in that state for a variable amount of time, t wait. Presumably, the electron in the dot will decay to the ground state with time. After the waiting period, the dot is moved to a measurement mode. Here, the gate potential is tuned so that only the spin-down state can tunnel out to the reservoir. Part a of Figure 3 shows the sequence of gate voltages in one cycle of the experiment. Part b shows the expected behavior of the change in current through the QPC, I QP C. Since the QPC is capacitatively coupled to the gate electrode, the change in I QP C follows the gate potential. However, it also depends on the the charge inside the dot. If the electron in the dot were in a spin-down state, then during the measurement step, the dot has electrons leading to a momentary increase in the the current through the QPC. This is the circled region in the figure. Part c of the figure demonstrates the experiment schematically. Figure 3 shows a sample run of the experiment. A small increase in I QP C can be seen for the spin-down state. As expected, the probability of measuring spin-up increased with t wait in this experiment. One major disadvantage of this readout scheme is that it requires a large magnetic field to produce a Zeeman splitting large enough to be detected. Another disadvantage is its sensitivity to fluctuations in the background charge. The next spin readout mechanism partly alleviates both these problems 5

27 Figure 7: Experimental setup for non-destructive measurement of spin. Figure from [6]. Figure 8: Typical data collected during the experiment in Figure 7. Figure from [6] Spin to Charge Conversion using Difference in Tunneling Rates of the Spin States This mechanism of spin readout utilizes a difference in tunnel rates between spin states. It has been observed that there is no significant difference in the tunneling rates between the spin states of a single electron. Hence, this mechanism does not work for a single electron. However, there is an appreciable difference between the tunneling rates of the electron singlet and triplet states [5]. The triplet state has a higher tunneling rate owing to the appreciable overlap of the antisymmetric orbital wavefunction of the spin triplet with that of electron states in the reservoir. In this mechanism, during measurement the gate potential is changed so that both the singlet and triplet states can tunnel into the reservoir. The presence of a singlet state is measured by noting that the quantum dot has electrons inside the dot for a short period of time even after the gate potential is changed owing to slow tunneling from the singlet state. Figure 5 shows the experimental setup [5]. As before, this is a 3-step process. Initialization is performed by injecting 1 electron into the dot. Then a second electron is allowed to enter the dot by making the S and T states energetically accessible. This is done for a waiting time t wait. Then the dot potential is brought back to the previous level executing a measurement. The sequence of gate voltage pulses is illustrated in part a of the figure. Part b shows the expected response of the QPC. The dip in I QP C after the gate voltage has been brought back corresponds to the time it took for an electron to tunnel out of the dot. Figure 6 shows typical results from actual runs. When the electrons are in a triplet state, tunneling is so fast that no dip in the QPC current is seen. However, for a singlet state, a clearly discernible dip can be seen Non-destructive Measurement of Spin The mechanism for a non-destructive measurement of spin is also based on the difference in tunneling rates of the S and T states. Suppose that the tunneling rate of the S state is characterized by a timescale t 1 and that of the T state is characterized by t. This mechanism is very similar to the previous one. As in Figure 7, the system is initialized with one electron. Then an additional electron is added and hence the system moves to either the T or S state. Then for measurement, a pulse of duration t is applied to the gate electrode such that it is energetically possible for an electron to tunnel out of both electron states. The key idea in this experiment is that t is set s.t. t 1 < t < t [6]. Therefore, during the pulse, an electron in the triplet state effectively tunnels out but one in the singlet state does not. Immediately subsequent to the pulse, if the dot has only 1 electron, another electron can tunnel into the dot. However, since tunneling into the triplet state is very fast, it is highly likely that this will happen. Therefore, at the end of the measurement, the dot has the same state as before. Consequently, this is a non-destructive measurement of spin. The triplet state is correlated with a momentary reduction in charge of the quantum dot. 6

28 Figure 9: Schematic of a QPC fabricated on a DEG. Figure from [6]. Figure 3: Conductance quantization in a QPC. Figure from [1]. Figure 8 shows experimental traces of the change in QPC current during this experiment with consecutive measurements. Each measurement results in a dip in I QP C because of capacitative coupling. If the measurement outcome is a triplet state, the dip is smaller in magnitude owing to the momentary reduction in charge of the dot. As expected, the number of trials with the same outcome for both measurements significantly exceeded the number of trials with different outcomes thereby demonstrating the non-destructive nature of this measurement. 8. Charge Measurement using QPCs All the spin readout mechanisms mentioned above use Quantum Point Contacts, QPCs, as electrometers. QPCs are devices in which current is constricted to flow through a very narrow region. Figure 9 shows a Quantum Point Contact defined by electrostatic potentials. A narrow constriction for electron flow from the left of the DEG to the right is created by applying negative voltages to the gates on the DEG. This results in the quantization of conductance to integer multiples of e h for current flow through the QPC at low temperatures. A plot of the conductance vs. potential difference across a QPC for small values of the potential difference can be see in Figure 3. Since the slope of the conductance is extremely high near e h, which is marked by a plus in the figure, the conductance is highly sensitive to electrostatic potentials nearby. In particular, the increase in the charge in a quantum dot next to a QPC reduces the conductance of the QPC due to electron-electron repulsion. The origin of conductance quantization can be understood with a simple model of a QPC. We assume that the QPC is an ideal and pure 1-dimensional structure with electron reservoirs at both ends [7]. These reservoirs are at chemical potentials E F and E F + δu. The wavevector of electrons through the QPC is quantized in dimensions and continuous in the third one. The energy of a particular state is: E(n, k) = E n + k Here k is the wavenumber for propagation through the QPC and n indexes the sub-band created due to confinement in the other directions. The number of occupied sub-bands, and hence the number of modes through which current can propagate, is the largest integer N, s.t. E N < E F. The group velocity of each sub-band is v n = de(n,k) dk and dk the density of states for a 1 dimensional particle in a box is ρ n = πde(n,k). The current through each sub-band is the product of the density of states and the group velocity. As seen from the expressions above, this is independent of n. Therefore, each sub-band carries the same amount of current, which is equal to ev n ρ n δu = e e π δu = h δu. Since current flows through an integer number of sub-bands, the total current is an integer multiple of e h δu, and hence the conductance is an integer multiple of e h. m (89) 7

29 9 Environmental Interactions Up until now, we have been discussing the ideal behavior of semiconductor quantum dots in the few-electron regime. Non-idealities are what makes quantum computing a difficult problem. Non-ideality is introduced by the interaction of quantum dot devices with their environments. Environmental interactions mainly serve to introduce decoherence which is the bane of quantum computing. For quantum computing to be more powerful that classical computing, the qubits must be coherent. Consider a quantum state that is a linear superposition of classical states. Coherence implies the preservation of the phase relationship through time. Decoherence of a linear superposition of two states is the same as converting it to classical mixture of states with some probability distribution on the states. Although environmental interaction and decoherence are extremely complicated phenomenon, major features are captured by two time scales, T 1 and T. 9.1 T 1 and T T 1 is the timescale involved in the change in occupancy of the difference quantum states. Consider a single spin in a static magnetic field. corresponds to a state in which the spin is pointing in the direction of the magnetic field and corresponds to a spin pointing in the opposite direction. Now consider a quantum superposition of the spins X = 1 ( + e iθ ). An ensemble of such spins involves an equal occupation of and states along with a definite phase relationship, θ, between them. Energy exchange with the environment may lead to the ensemble evolving in such a way that occupation of and states is no longer half and half. The time scale in which this happens is called T 1. In a Bloch sphere representation, this can be though of as the amount of time it takes for the spin to tip over from the North Pole to the equator. T is the timescale involved in the loss of phase coherence. If an ensemble of spins X evolve such that they no longer have a fixed phase relationship between them, i.e. they lose memory of θ, then the spins are said to lose phase coherence. In a Bloch Sphere, this corresponds to an ensemble of spins starting at a fixed latitude and longitude but ending up in all longitudes at that specific latitude. T is the time it takes for that to happen. Therefore, T 1 can be though of as the timescale involved in the movement of a spin ensemble along the longitude and hence the name longitudinal time. Similarly T can be thought of as the time required for a spin ensemble to spread itself along a given latitude and hence the name transverse time. In the density matrix view, an ensemble of spins with a classical probability distribution over them is considered. Consequently, a state of a system corresponds to its density matrix which can be thought of as a vector in the Bloch sphere with length 1. A coherent state, has length 1. T in this picture can be thought of as the time required for the Bloch vector to shrink its length to. For single spins localized in semiconductor quantum dots, T 1 is found to be quite long. It is on the order of a millisecond [1]. This is expected because of the small magnetic moment of an electron. Exchange of energy with the environment becomes difficult because of this and hence the long longitudinal lifetimes of spins. Very short T is the major problem in doing coherent quantum computation. In the next section, we describe an experiment that utilizes the very source of phase decoherence, nuclear spins, to increase T by several orders of magnitude []. 9. Use of spin-echo to increase the coherence time of coupled dots Due to the nuclear spins in GaAs, electrons in quantum dots experience an effectively random nuclear field, which changes over time. Therefore, the nuclear field experienced by the quantum dot changes across different runs of an experiment. However, within one run, for instance, one swap operation, the nuclear field remains constant. Petta et al. use this property of the short term constancy of the the nuclear field to increase T from a mere ns to more than 1µs []. The basic idea is to rotate the spin by exactly 18 degrees so that the nuclear field acts to rotate the spins in the opposite direction. This effectively undoes the decoherence induced by rotation of the spins due to the nuclear field. 8

30 Figure 31: Use of Spin Echo to increase coherence time of spins. A. Pulse sequence used during the experiment. B. Probability of the spin being in (, )S and the end of one cycle vs. detuning and exchange time, τ E. C. Probability of Singlet with a fixed τ E and varying τ S τ S for various values of τ S + τ S. Here ɛ is fixed. Figure from []. Their experiment is shown in Figure 31. After initialization into the (, )S state, ɛ is lowered rapidly which moves the system to (1, 1)S. Then the system is left to evolve for a variable amount of time τ S. During this period, the random nuclear field serves to rotate the spin along the x-axis of the Bloch Sphere. Then ɛ is made close to zero and hence J is large. This is done for a duration τ E. Then ɛ is again made very -ve letting the nuclear field rotate the spins again for time τ S. The key point is that τ E was such that the spin was rotated exactly π, 3π,... radians during the exchange pulse, i.e. J(ɛ)τ E / = π, 3π,.... Hence the nuclear field acts in the exactly opposite direction and ends up undoing the rotation around the x-axis. As can be seen from Part C of Figure 31,when τ S = τ S the probability of observing the spin singlet state is very high. Similarly, in part B, they show that the probability of seeing a spin singlet as a function of ɛ and τ E. Here τ S = τ S. Probability of seeing a spin singlet clearly increases when τ E is such that the rotation incurred is an odd multiple of π. 1 Conclusion In this paper, we discussed a possible implementation of quantum computers using spins of semiconductor quantum dots in this paper. We described and analyzed the basic physics of spins in quantum dots throughout the paper. Also, we described key experimental results in this field. The state-of-the-art in this proposal for quantum computing has already realized many of the key elements necessary of a scalable quantum computer. It is possible to reliably initialize the state of the spins and successfully manipulate both single and coupled spins. Furthermore, various mechanisms for spin readout are also available. The immediate problem that needs to addressed to make this proposal truly scalable is the lack of robustness of the elements described in the paper. The single spin rotation described in Section 6 only has a fidelity of.73 which means that the rotation is not accurate almost of quarter of the time. Similarly the fidelity of the U sw gate is also not 9