Quantum Dots

1. Quantum Dots By Timothy Paik Marcus Dahlstrom Michael Nip

2. Implementing Quantum Computers • Many implementations for quantum computing • Why solid state? • Scalability • Decoherence is less of a problem

3. What is a quantum dot? • In two words, a semiconductor nanocrystal. • Easily tunable by changing the size and composition of the nanocrystal

4. Gallium Arsenide Quantum Dots • Gallium arsenide is a III-V semiconductor • Higher saturated electron velocity and higher electron mobility than silicon • Gallium arsenide can emit and absorb light, unlike silicon • No silicon laser is possible (or has been made yet)

5. Energy Band Levels • Electrons exist in discrete energy levels in bulk semiconductor material. • There exists a forbidden range of energy levels in any material called the band gap.

6. Energy Band Levels • By absorbing some sort of stimulus (in light or heat form), an electron can rise to the conduction band from the valence band. • This action leaves behind a “hole” in the valence band. The hole and the electron together are called an exciton.

7. Energy Band Levels • The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius. • When the size of the semiconductor falls below the Bohr Radius, the semiconductor is called a quantum dot.

8. Tuning Quantum Dots • By changing size, shape, and composition, quantum dots can change their absorptive and emissive properties dramatically

9. Manufacturing methods • Electron beam lithography • Molecular beam epitaxy

10. Electron Beam Lithography • Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer • λ = (12.3 Å / √V) • Advantages: • generation of micron and submicron resist geometries • greater depth of focus than optical lithography • masks are unnecessary • Optical diffraction limit is not a real concern

11. Electron Beam Lithography • Disadvantage(s): • The lithography is serial (masks aren’t used; instead the beam itself sweeps across the wafer) => Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution • The proximity effect: Electrons scatter because they are relatively low in mass, reducing the resolution. • Heavy ion lithography has been proposed, but still is in development stages

12. Molecular Beam Epitaxy • Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystal • This is done by evaporating each of the elements to combine, then condensing them on top of the wafer. • The word “beam” means that the evaporated atoms only meet each other on the wafer

13. Spin Quantum Computing Qubit information is stored in the spin state of an electron in an artificial atom Advantages: Long decoherence time Future Scalabilty Artifical atoms are bigger than regular atoms therefore easier to manipulate

14. Decoherence time ~ 100ns • Time before the quantum mechanical system starts acting in a classical way with it's complex environment • The state of the system has not yet collapsed due to (unwanted) environmental effects • Spin - DT are 100 as long as for the Exciton • Need to SWITCH 104 during DT for reliable error correction. This requirement is met.

15. Artificial Atom • Double Barrier Heterostructure • Dot: In0.05Ga0.95As • Source &Drain : GaAs • 2D Electron Gas • Confine with gate bias • D ~ Fermi wavelength → Discrete energy levels

16. Adding Electrons, changing Vgate • 2D-Harmonic Oscillator • Shell structure as in atoms • Magic Numbers: 2, 6, 12... • To add “even” electron requires only additional Coulomb energy

17. Hydrogen:Energy levels ~ 1eVSize ~ 1Å Only strong magnetic fields can perturb energy levelsFactor 1000... Comparison with Hydrogen • Artificial Atom:Energy levels ~ 1meVSize ~ 10μmWeak magnetic fields can affect energy levels

18. Tuning the Quantum Dot • Tune so we have one valence electron • Initial state can be set by applying homogeneous magnetic field → |0> • Low temperature: kT < ΔE (state gap) • Now we have defined our single qubit Energy Unoccupied state Gate bias Spin up - electron position

19. Single Qubit manipulation • Unitary operations can be made by applying a local magnetic field: HZE = -μ·B = g μB S·B • MF microscope • AF microscope • Sub grid of current • Magnetic dots • Etc... (Magnetic force microscope tip)

20. Two Qubit Manipulation • Complete set of logic requires a CNOT • Dots are placed so close that they overlap and interact: • Hspin = J(t)S1·S2Exchange coupling:J(t,E,B) = Etriplet -Esinglet (4:th order Harmonic Oscillator)

21. Ground State Splitting (J = Et – Es) • 2 coupled fermions must have an total anti-symmetric wave function • Lowest coupled state is the singlet. It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates):|ψs> ~ (|12> + |21>) (|↓↑> - |↑↓>) • The triplet states are: (|↓↓>)|ψt> ~ (|12> - |21>) (|↓↑> + |↑↓>) (|↑↑>) • <1|2> ≠ 0, |i> is spatial w.f. Coulomb dominates

22. Solving J(B(t)): Exchange Coupling • Different solutions:* Heitler-London* Hund-Mulliken * Hubbard • Important conclusion:We can control coupling from zero to non-zero by changing the magnetic field → We can perform two qubit operations.

23. SWAP - gate • Assume J can be pulsed:J(t) = {0, J0}Formula 1Formula 2 • Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

24. XOR ~ CNOT • Formula 3 • Requirements: * Spin rotations about the z-axis * Squareroot of Uswap

25. Read out / Memory • Assume dot with an electron with some information stored in spin-state • Connect two leads to dot • Apply a small bias (ΔV) → Current (i)?! Energy Unoccupied state i? Gate bias Spin up - electron position

26. Another Spin up electron enters dot • Pauli principle forces electrons with spin up to occupy the higher energy state • Negligible chance of tunneling i=0 E Higher energy level (forbidden classically) Gate bias Spin up - electron position

27. Spin down electron enters dot • Pauli principle allows the new electron to join the same energy level as the original electron • Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow E Unoccupied state i≠0 Gate bias Spin up - electron position

28. Read out / Memory • We have a way of measuring the spin state of an electron in a quantum dot • The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties • To be able to predict the original state of the dot, the state has to be prepared again and then measured using the same technique • The electron current can be specialized (we can aim it's spin to make measurement efficient)

29. 5 DiVincenzo QC Criteria • A scalable physical system with well-characterized qubits. • The ability to initialize the state of the qubits to a simple fiducial state. • Relatively long decoherence times compared to gate-operation times. • A universal set of quantum gates. • Qubit-specific measurement capability.

30. The Physical System: Excitons Trapped in GaAs Quantum Dots • Exciton - a Coulomb correlated electron-hole pair in a semiconductor, a quasiparticle of a solid. • Often formed when photons excite electrons from the valence band into the conduction band. • Wavefunctions are “hydrogen-like” i.e. an “exotic atom” though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses • Decay by radiating photons. Decay time ~50ps-1ns • Hence can define the computational basis as absence of an exciton |0>, or existence of an exciton |1>

31. Initialization • Register relaxes to the |00…0> state within 50ps-1ns due to radiative decay • Experimental systems are cooled to liquid helium temps ~4K to prevent thermal excitations • Hence initialization with such a system is relatively easy • Other states can be initialized by applying gates to the register

32. Relatively Long Decoherence Times • Mechanisms: • Radiative Decay ~10ps-1ns • Can be lengthened by electron-hole separation • Background Electromagnetic fluctuations • Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral. • Gate times are determined by energy band spacing, i.e. creation and annihilation energies. • Gate operations for GaAs QDs are estimated at ~1ps or less

33. A Universal set of Quantum Gates • Single Qubit Rotations through laser induced Rabi Oscillations • CNOT operations through dipole interactions and laser excitation

34. Single Qubit Gates: Rabi Flopping • Light-particle interaction is characterized by the product of the dipole moment and the electric field: μ•E(t)= ħR(t) Where R(t) is the Rabi frequency and the pulse area is given by: Θ(t)=∫R(t)dt and the state at time t is then given by: Cos(Θ/2)|0>+Sin(Θ/2)|1>

35. Stufler et al. Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses. Cos(Θ/2)|0>+Sin(Θ/2)|1> |1> => electric charge =>Photocurrent (PC) PC~Sin2(Θ/2) π-pulse corresponds to a population inversion

36. CNOT: Dipole Coupling Nearest neighbor interactions alter the energy states: Effective energy: E’i = Ei + ∑j≠i ∆Eij nj Hence, a coherent π-pulse with energy E’t(nc) results in a state flop iff the control state is occupied.

37. Electrostatic Dipole fields fall off as 1/R^3 hence the CNOT gate can only be used for closely neighboring QDs. Solution: Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots. Optical Cavity coupling and fiber optical interconnects have also been proposed. Overcoming Short Interaction Distances

38. Read Out of Specified Qubit States • Optical readout: Excitons decay spontaneously and the resulting radiation can be measured. Alternatively, an excitation/probe beam spot can be physically positioned in the region of the desired QD. Due to the statistical distribution of QD shape and size variations, individual QDs can be more accurately identified and addressed through frequency discrimination. In either case, repeated measurements have to be made. A single shot readout still needs to be developed.

39. 5 DiVincenzo QC Criteria • A scalable physical system with well-characterized qubits. • The ability to initialize the state of the qubits to a simple fiducial state. • Relatively long decoherence times compared to gate-operation times. • A universal set of quantum gates. • Qubit-specific measurement capability.