Quantum Dots

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Transcript of Quantum Dots

  • Quantum DotsBy Timothy PaikMarcus DahlstromMichael Nip

  • Implementing Quantum ComputersMany implementations for quantum computing

    Why solid state?ScalabilityDecoherence is less of a problem

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

  • Gallium Arsenide Quantum DotsGallium arsenide is a III-V semiconductorHigher saturated electron velocity and higher electron mobility than siliconGallium arsenide can emit and absorb light, unlike siliconNo silicon laser is possible (or has been made yet)

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

  • Energy Band LevelsBy 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.

  • Energy Band LevelsThe 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.

  • Tuning Quantum DotsBy changing size, shape, and composition, quantum dots can change their absorptive and emissive properties dramatically

  • Manufacturing methodsElectron beam lithographyMolecular beam epitaxy

  • Electron Beam LithographyElectrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer = (12.3 / V)Advantages:generation of micron and submicron resist geometriesgreater depth of focus than optical lithographymasks are unnecessaryOptical diffraction limit is not a real concern

  • Electron Beam LithographyDisadvantage(s):The lithography is serial (masks arent used; instead the beam itself sweeps across the wafer) => Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolutionThe 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

  • Molecular Beam EpitaxyMolecular 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 crystalThis 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

  • Spin Quantum ComputingQubit 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

  • Decoherence time ~ 100nsTime before the quantum mechanical system starts acting in a classical way with it's complex environmentThe state of the system has not yet collapsed due to (unwanted) environmental effectsSpin - DT are 100 as long as for the ExcitonNeed to SWITCH 104 during DT for reliable error correction. This requirement is met.

  • Artificial AtomDouble Barrier HeterostructureDot: In0.05Ga0.95AsSource &Drain : GaAs2D Electron GasConfine with gate biasD ~ Fermi wavelength Discrete energy levels

  • Adding Electrons, changing Vgate2D-Harmonic OscillatorShell structure as in atomsMagic Numbers: 2, 6, 12...To add even electron requires only additional Coulomb energy

  • Comparison with HydrogenArtificial Atom: Energy levels ~ 1meV Size ~ 10m Weak magnetic fields can affect energy levelsHydrogen: Energy levels ~ 1eV Size ~ 1 Only strong magnetic fields can perturb energy levels Factor 1000...

  • Tuning the Quantum DotTune so we have one valence electronInitial state can be set by applying homogeneous magnetic field |0>Low temperature: kT < E (state gap)Now we have defined our single qubitEnergypositionGate biasSpin up - electronUnoccupied state

  • Single Qubit manipulationUnitary operations can be made by applying a local magnetic field: HZE = -B = g B SB MF microscopeAF microscopeSub grid of currentMagnetic dotsEtc...(Magnetic force microscope tip)

  • Two Qubit ManipulationComplete set of logic requires a CNOTDots are placed so close that they overlap and interact:Hspin = J(t)S1S2 Exchange coupling: J(t,E,B) = Etriplet -Esinglet (4:th order Harmonic Oscillator)

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

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

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

  • XOR ~ CNOTFormula 3 Requirements: * Spin rotations about the z-axis * Squareroot of Uswap

  • Read out / MemoryAssume dot with an electron with some information stored in spin-stateConnect two leads to dotApply a small bias (V) Current (i)?!EnergypositionGate biasSpin up - electronUnoccupied statei?

  • Another Spin up electron enters dotPauli principle forces electrons with spin up to occupy the higher energy stateNegligible chance of tunnelingEpositionGate biasSpin up - electronHigher energy level(forbidden classically)i=0

  • Spin down electron enters dotPauli principle allows the new electron to join the same energy level as the original electronCoulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flowEpositionGate biasSpin up - electronUnoccupied statei0

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

  • 5 DiVincenzo QC CriteriaA 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.

  • The Physical System: Excitons Trapped in GaAs Quantum DotsExciton - 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 massesDecay by radiating photons. Decay time ~50ps-1nsHence can define the computational basis as absence of an exciton |0>, or existence of an exciton |1>

  • InitializationRegister relaxes to the |000> state within 50ps-1ns due to radiative decayExperimental systems are cooled to liquid helium temps ~4K to prevent thermal excitationsHence initialization with such a system is relatively easyOther states can be initialized by applying gates to the register

  • Relatively Long Decoherence TimesMechanisms:Radiative Decay ~10ps-1nsCan be lengthened by electron-hole separationBackground Electromagnetic fluctuationsLess 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

  • A Universal set of Quantum GatesSingle Qubit Rotations through laser induced Rabi OscillationsCNOT operations through dipole interactions and laser excitation

  • Single Qubit Gates: Rabi FloppingLight-particle interaction is characterized by the product of the dipole moment and the electric field:

    E(t)= R(t)

    Where R(t) is the Rabi frequency and the pulse area is given by:


    and the state at time t is then given by:Cos(/2)|0>+Sin(/2)|1>

  • Stufler et al.Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses.


    |1> => electric charge

    =>Photocurrent (PC)PC~Sin2(/2)

    -pulse corresponds to a population inversion

  • CNOT: Dipole CouplingNearest neighbor interactions alter the energy states:

    Effective energy: Ei = Ei + ji Eij nj

    Hence, a coherent -pulse with energy Et(nc) results in a state flop iff the control state is occupied.

  • Overcoming Short Interaction DistancesElectrostatic 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